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Determining the total variation mixing time of Kac's random walk on thespecial orthogonal group $\mathrm{SO}(n)$ has been a long-standing openproblem. In this paper, we construct a novel non-Markovian coupling forbounding this mixing time. The analysis of our coupling entails controlling thesmallest singular value of a certain random matrix with highly dependententries. The dependence of the entries in our matrix makes it not-amenable toexisting techniques in random matrix theory. To circumvent this difficulty, weextend some recent bounds on the smallest singular values of matrices withindependent entries to our setting. These bounds imply that the mixing time ofKac's walk on the group $\mathrm{SO}(n)$ is between $C_{1} n^{2}$ and $C_{2}n^{4} \log(n)$ for some explicit constants $0 < C_{1}, C_{2} < \infty$,substantially improving on the bound of $O(n^{5} \log(n)^{2})$ by Jiang. Ourmethods may also be applied to other high dimensional Gibbs samplers withconstraints and thus are of independent interest. In addition to givinganalytical bounds on the mixing time, our approach allows us to computerigorous estimates of the mixing time by simulating the eigenvalues of a randommatrix.

On the Mixing Time of Kac's Walk and Other High-Dimensional Gibbs Samplers with Constraints